Mathematics for the interested outsider

Conservative Vector Fields

For a moment, let’s return to the case of Riemannian manifolds; the vector field analogue of an exact -form is called a “conservative” vector field , which is the gradient of some function .

Now, “conservative” is not meant in any political sense. To the contrary: integration is easy with conservative vector fields. Indeed, if we have a curve that starts at a point and ends at a point then fundamental theorem of line integrals makes it easy to calculate:

I didn’t go into this before, but the really interesting thing here is that this means that line integrals of conservative vector fields are independent of the path we integrate along. As a special case, the integral around any closed curve — where — is automatically zero. The application of such line integrals to calculating the change of energy of a point moving through a field of force explains the term “conservative”; the line integral gives the change of energy, and whenever we return to our starting point energy is unchanged — “conserved” — by a conservative force field.

This suggests that it might actually be more appropriate to say that a vector field is conservative if it satisfies this condition on closed loops; I say that this is actually the same thing as our original definition. That is, a vector field is conservative — the gradient of some function — if and only if its line integral around any closed curve is automatically zero.

As a first step back the other way, it’s easy to see that this condition implies path-independence: if and go between the same two points — if and — then

Indeed, the formal sum is a closed curve, since , and so

Of course, this also gives rise to a parallel — and equivalent — assertion about -forms: if the integral of around any closed -chain is always zero, then for some function . Since we can state this even in the general, non-Riemannian case, we will prove this one instead.

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